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Compactly supported tight and sibling frames with maximum vanishing moments. (English) Zbl 1016.42023
Two families \[ \Psi = \{ \psi_{i,j,k} = 2^{j/2} \psi_i (2^j\cdot - k): 1 \leq i \leq N, j,k \in {\mathbf Z} \} \] and \[ \tilde \Psi = \{ \tilde\psi_{i,j,k} = 2^{j/2} \tilde\psi_i (2^j\cdot - k): 1 \leq i \leq N, j,k \in {\mathbf Z} \} \] are called sibling frames if the frame generators \(\psi_i, \tilde\psi_i, i=1, \ldots, N\), are generated by the same refinable function \(\phi\), if they are Bessel families, and if the duality relation \(\langle f,g \rangle = \sum_{i=1}^N \sum_{j,k \in {\mathbb{Z}}} \langle f, \psi_{i,j,k} \rangle \langle \tilde\psi_{i,j,k}, g \rangle\) holds for all \(f,g \in L^2({\mathbb{R}})\).
One of the main results of the paper under review is that there exist two compactly supported sibling frames with the maximal number of vanishing moments, which can be chosen to be symmetric or antisymmetric. The proof of this result is constructive. The authors also provide the characterization of sibling frames in terms of the vanishing moment recovery function, which is then used in the construction of a tight frame for \(L^2({\mathbf R})\) with two compactly supported generators that both have the maximal number of vanishing moments. This tight frame does not have symmetry or antisymmetry.
The results obtained in this paper are related to the work of I. Daubechies, B. Han, A. Ron and Z. Shen [“Framelets: MRA-based constructions of wavelet frames”, Appl. Comput. Harmon. Anal. 14, No. 1, 1-46 (2003; Zbl 1035.42031)].

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15 Spline approximation
Software:
DT-CWT
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